Molecular diffusion in the laminar sub-layer during turbulent flow in a smooth tube

2004 ◽  
Vol 59 (6) ◽  
pp. 1191-1197 ◽  
Author(s):  
C.R. Huang ◽  
A.F. Denny ◽  
N.W. Loney
Keyword(s):  
Turbulent Flow ◽  
Molecular Diffusion ◽  
Smooth Tube

Turbulent Dispersion in Porous Materials as Modeled by a Mixing Cell with Stagnant Zone

1971 ◽  
Vol 11 (01) ◽  
pp. 57-62
Author(s):  
C.R. Kyle ◽  
R.L. Perrine
Keyword(s):  
Porous Material ◽  
Turbulent Flow ◽  
Porous Materials ◽  
Molecular Diffusion ◽  
Stagnant Zone ◽  
Skewed Distribution ◽  
Capillary Tubes ◽  
Fluid Movement ◽  
The Media ◽  
Flow Through

Abstract This paper reports on a simple theoretical analysis of dispersion in rapid flow through porous materials, giving a comparison of predicted results with experiments. The analytical model considers a pore structure which acts like a sequence of mixing cells, each coupled with a stagnant zone. Computed results compare very favorably with experimental observations on flow through a staggered matrix of cylinders. This, in turn, has been shown to behave the packed beds of spheres with corresponding properties. Agreement requires that values for certain theoretical parameters be fitted from the data The values required for these parameters are very reasonable. Development of parameters are very reasonable. Development of this approach could be useful for a number of related problems. Introduction The dispersion of two dynamically similar miscible liquids in laminar or turbulent flow through a porous material is a very complex process. However, it can be broken down into four process. However, it can be broken down into four basic mixing mechanisms:Molecular diffusion. Where the flow velocity is appreciable, or pore size is larger, diffusion is usually negligible. Molecular diffusion will not be discussed in this paper.Uneven fluid movement due to irregular pore geometry and inhomogeneities in the media. Both of these factors are difficult to treat, and are usually neglected in theoretical analysis.Uneven fluid movement due to velocity differences within the pores and passages. The zero-velocity boundary condition on each solid surface assures this type of mixing in both laminar and turbulent flow.Mixing by rotational flow, or by turbulent eddies within the pores or passages. The last two are both convective mixing processes and depend primarily upon the level of processes and depend primarily upon the level of energy dissipation in the media, as well as on the geometry of the system. In general as the velocity increases and the friction losses rise, so does the efficiency of the mixing process. Dispersion has been reviewed thoroughly by Perkins and Johnston and has been studied Perkins and Johnston and has been studied extensively by others. DIFFUSION MODEL OF DISPERSION The most commonly used mathematical model for dispersion in both laminar and turbulent flow is a diffusion-type equation (Refs. 1 or 5). The solution for a step function input with flow in the x-direction only, and with negligible lateral gradients, shows that an initial sharp interface degenerates into a broad mixing zone which grows approximately as the square root of the distance traveled. The solution also predicts a normal distribution for concentration as a function of distance. However, in most real systems "tailing" occurs, causing a skewed distribution. Usually the deviation is not serious and the diffusion equation may be used as a good approximation for the actual process. process. DISPERSION IN A TUBE Another simple model for laminar dispersion, neglecting molecular diffusion, is to consider a porous material as a bundle of capillary tubes. porous material as a bundle of capillary tubes. Sir Geoffrey Taylor showed that if one fluid in a capillary tube is displaced by another dynamically similar miscible fluid, the average concentration, C, at the tube exit is given by: 2C = (V /2V)p SPEJ P. 57

A Numerical Study of Turbulent Flow in Helical Static Mixers

2006 ◽  
Author(s):  
Ramin K. Rahmani ◽  
Anahita Ayasoufi ◽  
Theo G. Keith
Keyword(s):  
Turbulent Flow ◽  
Molecular Diffusion ◽  
Stokes Equations ◽  
Numerical Study ◽  
Critical Role ◽  
Three Dimensional ◽  
Turbulent Dispersion ◽  
Working Fluid ◽  
Static Mixer ◽  
Static Mixers

Many processing applications call for the addition of small quantities of chemicals to working fluid. Hence, fluid mixing plays a critical role in the success or failure of these processes. An optimal combination of turbulent dispersion down to eddies of the Kolmogoroff scale and molecular diffusion would yield fast mixing on a molecular scale which in turn favors the desired reactions. Helical static mixers can be used for those applications. The range of practical flow Reynolds numbers for these mixers in industry is usually from very small (Re ∼ 0) to moderate values (Re ∼ 5000). In this study, a helical static mixer is investigated numerically using Lagrangian methods to characterize mixer performance under turbulent flow regime conditions. A numerical simulation of turbulent flows in helical static mixers is employed. The model solves the three-dimensional, Reynolds-averaged Navier-Stokes equations, closed with the Spalart-Allmaras turbulence model, using a second-order-accurate finite-volume numerical method. Numerical simulations are carried out for a six-element mixer, and the computed results are analyzed to elucidate the complex, three-dimensional features of the flow. Using a variety of predictive tools, mixing results are obtained and the performance of static mixer under turbulent flow condition is studied.

The diffusion behind a line source in homogeneous turbulence

1954 ◽  
Vol 224 (1159) ◽  
pp. 487-512 ◽  
Keyword(s):  
Turbulent Flow ◽  
Decay Time ◽  
Molecular Diffusion ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Particle Diffusion ◽  
Velocity Correlation ◽  
Rate Of Spread ◽  
Heated Wire ◽  
The Mean

The theory of turbulent diffusion by continuous movements relates the mean particle diffusion from a fixed source to the Lagrangian velocity correlation function, and measure­ments of the diffusion of heat behind a thin heated wire in a uniform turbulent flow have been used to compute this correlation, assuming that the processes of diffusion by con­tinuous movements and molecular conduction are statistically independent. A series of measurements both of the mean temperatures and the temperature fluctuations in the wake of a thin heated wire has been made in the uniform turbulent flow behind bi-plane grids, for grid Reynolds numbers between 2700 and 21000 and within the initial period of decay of the turbulence. In these measurements, the rate of spread of the heat wake was determined in two ways, directly from measurements of the turbulent transport of heat and by numerical differentiation of widths computed from observations of mean temperature. The extent of the accelerated diffusion of heat, which is caused by intensification of the temperature gradients by the turbulent motion, can be computed from the measurements of lateral temperature correlations in the flow, and was found to be comparable with the total diffusion. Both the total diffusion process and the process of accelerated molecular dif­fusion are very nearly self-preserving during decay in the initial period, with a time scale that varies as the decay time and a velocity scale varying inversely as the square root of the decay time, which is consistent with the observed self-preservation of Eulerian correlations. The rate of spread of the heat wake is simply related to the particle diffusion only for short diffusion times at ordinary Reynolds numbers, and the mean-square particle accelera­tion can be computed. The results are significantly larger than those found by other workers who have neglected the additional spread of the wake by accelerated molecular diffusion.

scholarly journals Some remarks on modelling the PDF of the concentration of a dispersing scalar in turbulence

2002 ◽  
Vol 13 (1) ◽  
pp. 95-108 ◽  
Author(s):  
P. C. CHATWIN
Keyword(s):  
Probability Density Function ◽  
Evolution Equation ◽  
Turbulent Flow ◽  
Probability Density ◽  
Density Function ◽  
Molecular Diffusion ◽  
Concentration Field ◽  
Similarity Solutions ◽  
Small Scale ◽  
Probability Density Function Pdf

The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration uctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.

An Introduction to Turbulent Flow

2000 ◽  
Author(s):  
Jean Mathieu ◽  
Julian Scott
Keyword(s):  
Turbulent Flow

Modal Analysis of Turbulent Flow near an Inclined Bank–Longitudinal Structure Junction

2021 ◽  
Vol 147 (3) ◽  
pp. 04020100
Author(s):  
Nasser Heydari ◽  
Panayiotis Diplas ◽  
J. Nathan Kutz ◽  
Soheil Sadeghi Eshkevari
Keyword(s):  
Turbulent Flow ◽  
Modal Analysis ◽  
Longitudinal Structure
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